Localisation in buckling and post-buckling of cylindrical shells

Buckling of axially compressed cylinders is a classic problem in engineering mechanics. Even though an analytical solution based on a linear eigenvalue analysis has existed for over 100 years, the buckling of cylinders continues to attract attention from researchers. This interest stems from the unstable nature of the buckling event with its associated severe sensitivity to initial imperfections—geometric, loading, boundary conditions, or otherwise—that rapidly erode the analytical or numerical predictions based on the perfect problem. A classical analysis of the compressed cylinder based on a linear eigenvalue analysis also suggests a periodic nature of the buckling modes, whereas high-speed photography experiments indicate that buckling is governed by the formation of one or multiple dimples that then multiply to cover the whole cylinder surface. Asymptotic expansions around the critical point, and branch-switching techniques based on extended arc-length methods, show that the periodic buckling modes branching from the pre-buckling path do indeed localise immediately after the bifurcation into solutions featuring one or multiple dimples. Tracing these dimple solutions in a path-following solver with respect to applied compression demonstrates a sequential pattern formation whereby isolated dimples multiply circumferentially through a series of de- and restabilisations. Furthermore, the single-dimple solution forms an unstable equilibrium path, almost coincident with the pre-buckling path, that corresponds to the smallest energy barrier between the pre-buckling and post-buckling regimes. The small energy barrier associated with the single-dimple solution means that the compressed, pre-buckled cylinder is exceedingly sensitive to perturbations once the level of compression is exceeded for which the single dimple exists as an unstable equilibrium. It is possible to parametrise the compressive onset of the single-dimple solution using a single non-dimensional parameter, and the ensuing relation forms an alternative lower-bound design curve that shows good correlation with experimental results in the literature and other lower-bound curves suggested recently. The fact that localisations can form as unstable equilibrium solutions anywhere across the domain of the cylinder implies a large set of possible trajectories to instability, with each trajectory affine to a particular imperfection signature. This multiplicity of possible routes to buckling leads to a large spread in buckling loads even for seemingly indistinguishable random imperfections of equal amplitude. It is shown that the ability to control the equilibrium trajectory to buckling via dominant imperfections, or elastic tailoring using tow-steered composites, creates interesting possibilities for designing imperfection-insensitive shells.